First of all you need a hosting space. Your provider has certainly given you an FTP account name and password (FTP = file transfer protocol, that's the way you place your files on the server).
You need an FTP application, there are several free ones that work fine: Cyberduck (for Mac only), Filezilla, or FireFTP (an add-on for firefox). Be careful that you keep the directory structure for your files and pictures. Make sure that the "first page" of your site is named "index.html" (name it index in Quark). You need to include also the images, actually everything that Quark creates when you export your project to html.
Regarding fonts, you should of course stick with fonts that are well represented on the main platforms. There are some studies that establish the "market share" of common fonts, for example this survey.
PS: you can (and should!) define your custom font stacks under Edit > CSS Font Families .
Manuel S.
Technical Support Representative